Question

Curves in the graph shown give, as functions of radial distance $r$, the magnitude $B$ of the magnetic field inside and outside for long wires $a,b,c$ and $d$, carrying currents that are uniformly distributed across the cross-sections of the wires. The wires are far from one another. The current density in wire $a$ is

• A. greater than in wire $c$
• B. less than in wire $b$
• C. equal to that in wire $d$
• D. less than in wire $d$

Answer 1

The correct option is A greater than in wire $c$

Magnetic field at distance $r$ inside a wire from its axis is,

$B=\frac{{\mu }_{0}I}{2\pi R^2}r$

$\Rightarrow \frac{dB}{dr}=\frac{{\mu }_0}{2}\left(\frac{I}{\pi R^2}\right)$

Here, current density, $J=\frac{I}{\pi R^2}$

$\Rightarrow \frac{dB}{dr}\propto J$

$\therefore$ Slope of the $B-r$ curve $\propto J$

From the graph,

${\left[\frac{dB}{dr}\right]}_a>{\left[\frac{dB}{dr}\right]}_b>{\left[\frac{dB}{dr}\right]}_c$

$\therefore J_a>J_c$

Hence, option $\left(A\right)$ is the correct answer.